Divisors of the number of Latin rectangles

A k × n Latin rectangle on the symbols { 1 , 2 , … , n } is called reduced if the first row is ( 1 , 2 , … , n ) and the first column is ( 1 , 2 , … , k ) T . Let R k , n be the number of reduced k × n Latin rectangles and m = ⌊ n / 2 ⌋ . We prove several results giving divisors of R k , n . For example, ( k − 1 ) ! divides R k , n when k ⩽ m and m! divides R k , n when m < k ⩽ n . We establish a recurrence which determines the congruence class of R k , n ( mod t ) for a range of different t. We use this to show that R k , n ≡ ( ( − 1 ) k − 1 ( k − 1 ) ! ) n − 1 ( mod n ) . In particular, this means that if n is prime, then R k , n ≡ 1 ( mod n ) for 1 ⩽ k ⩽ n and if n is composite then R k , n ≡ 0 ( mod n ) if and only if k is larger than the greatest prime divisor of n.